How to Factor Polynomials — Step-by-Step Guide

Factoring polynomials feels like guesswork until you learn the pattern. Once you see them, factoring becomes a quick 4-step process. Here\u2019s how I teach it.

Step 1 — Always pull out the GCF first

Look at every term and find the Greatest Common Factor — the largest thing they all share. Pull it out front.

Example: 6x³ + 9x² - 12x → pull out 3x → 3x(2x² + 3x - 4)

Step 2 — Count the terms

• 2 terms? Try difference of squares or sum/difference of cubes
• 3 terms? It\u2019s a quadratic — use the AC method or trial and error
• 4 terms? Try factor by grouping

Step 3 — Apply the right pattern

Difference of squares (2 terms)

a² - b² = (a + b)(a - b)

Example: x² - 25 = (x + 5)(x - 5)

Quadratic trinomial (3 terms)

x² + bx + c — find two numbers that multiply to c and add to b.

Example: x² + 7x + 12 → 3 and 4 multiply to 12 and add to 7 → (x + 3)(x + 4)

Factor by grouping (4 terms)

Group the first two and last two terms, factor each group, then look for a common binomial.

Example: x³ + 2x² + 3x + 6 → x²(x + 2) + 3(x + 2) → (x + 2)(x² + 3)

Step 4 — Always check by multiplying back

If you can FOIL or distribute back to the original, you factored correctly. This catches 90% of mistakes.

Common mistakes I see in tutoring

• Skipping the GCF — makes the trinomial much harder
• Forgetting that signs matter (the 12 in x² + 7x + 12 means both factors are positive)
• Not checking — easy way to verify, often skipped

Practice tools

• Desmos graphing — factor visually by seeing where the graph crosses x-axis
• Wolfram Alpha — checks your work step-by-step
• Algebra tutoring — when patterns aren\u2019t clicking